Iterative solution of SPARK methods applied to DAEs

Transcription

1 Numerical Algorithms 31: Kluwer Academic Publishers Printed in the Netherlands Iterative solution of SPARK methods applied to DAEs Laurent O Jay Department of Mathematics 14 MacLean Hall The University of Iowa Iowa City IA USA ljay@mathuiowaedu; naljay@na-netornlgov Received 8 January 2001; revised 4 October 2001 In this article a broad class of systems of implicit differential algebraic equations DAEs is considered including the equations of mechanical systems with holonomic and nonholonomic constraints Solutions to these DAEs can be approximated numerically by applying a class of super partitioned additive Runge Kutta SPARK methods Several properties of the SPARK coefficients satisfied by the family of Lobatto IIIA-B-C-C -D coefficients are crucial to deal properly with the presence of constraints and algebraic variables A main difficulty for an efficient implementation of these methods lies in the numerical solution of the resulting systems of nonlinear equations Inexact modified Newton iterations can be used to solve these systems Linear systems of the modified Newton method can be solved approximately with a preconditioned linear iterative method Preconditioners can be obtained after certain transformations to the systems of nonlinear and linear equations These transformations rely heavily on specific properties of the SPARK coefficients A new truly parallelizable preconditioner is presented Keywords: differential algebraic equations holonomic constraints inexact modified Newton method Lobatto coefficients mechanical systems nonholonomic constraints overdetermined DAEs perturbation index preconditioning Runge Kutta methods stiffness AMS subject classification: 65F10 65H10 65L05 65L06 65L80 70F20 70F25 70H03 70H45 1 Introduction In this article a broad class of systems of possibly stiff and implicit differential algebraic equations DAEs is considered including Hessenberg DAEs of index 1 2 and 3 [15689] These equations encompass the formulation of mechanical systems with mixed constraints of holonomic nonholonomic scleronomic and rheonomic types [71617] Solutions to these DAEs can be approximated numerically by applying a class of super partitioned additive Runge Kutta SPARK methods such as the combination of Lobatto IIIA-B-C-C -D methods [9] SPARK methods can take advantage of splitting the differential equations into different terms and of partitioning the variables into different classes Several properties of the SPARK coefficients satisfied by the Lo- This material is based upon work supported by the National Science Foundation under Grant No This research was also supported in part by NASA Award No JPL subcontract No

2 172 LO Jay / Iterative solution of SPARK methods batto family are essential to treat the constraints and the algebraic variables properly A main difficulty for an efficient implementation of these methods lies in the numerical solution of the resulting systems of nonlinear equations For this purpose inexact modified Newton iterations can be used extending techniques proposed for the solution of implicit Runge Kutta IRK methods applied to implicit systems of stiff ordinary differential equations ODEs [10 12] Such an extension is not straightforward as the presence of constraints and algebraic variables adds some extra difficulty Linear systems of the modified Newton method can be solved approximately with a preconditioned linear iterative method after certain transformations to the systems of nonlinear and linear equations These transformations rely heavily on specific properties of the SPARK coefficients For an s-stage SPARK method and stiff DAEs the decomposition of at most s + 1 independent submatrices of the same dimension as the DAEs is required to build an efficient preconditioner The main purpose of this paper is to present the steps involved to put the system of linear equations of the modified Newton method in a form such that preconditioning these linear equations can be done in a straightforward manner following the results of [10 12] In section 2 the class of implicit DAEs considered in this article is presented In section 3 the definition of SPARK methods applied to these DAEs is given Some properties of the SPARK coefficients are given which are crucial for an efficient implementation of these methods In section 4 an approximate Jacobian to the system of nonlinear equations is derived after certain linear transformations Section 5 describes the steps involved to transform the approximate Jacobian before application of a preconditioner A new truly parallelizable preconditioner is succinctly presented 2 The system of implicit differential algebraic equations Consider the following class of systems of implicit differential algebraic equations DAEs d qty =vtyz dt 1a d pt y z = ftyzuγλψ dt 1b d ct y z u = dtyzuγλψ dt 1c mt y z u γ = 0 ht y z = 0 gty = 0 which may present some stiffness These equations encompass Hessenberg DAEs of index 1 2 and 3 [15689] They also include the formulation of mechanical systems with mixed constraints of holonomic nonholonomic scleronomic and rheonomic types [ ] In mechanics the quantities qvp represent respectively gener- 1d 1e 1f

3 LO Jay / Iterative solution of SPARK methods 173 alized coordinates generalized velocities and generalized momenta The right-hand side f of 1b contains generalized forces acting on the system and 1c describes the dynamics of external variables u The algebraic variables λ and ψ are Lagrange multipliers associated respectively to the nonholonomic constraints 1e and the holonomic constraints 1f The equations of constrained systems in mechanics can be derived from Newton s law of motion and the generalized Gauss variational principle of least constraint [13] It is assumed that the constraints 1f can also be expressed as gty = rtqty = 0 1g The variable t R is the independent variable and y = y 1 y n y T R n y z= z 1 z n z T R n z u= u 1 u n u T R n u γ = γ 1 γ n γ T R n γ λ= λ 1 λ n λ T R n λ ψ= ψ 1 ψ n ψ T R n ψ q : R R n y R n y p : R R n y R n z R n z c : R R n y R n z R n u R n u m : R R n y R n z R n u R n γ R n γ h : R R n y R n z R n λ g : R R n y R n ψ r : R R n y R n ψ v : R R n y R n z R n y f : R R n y R n z R n u R n γ R n λ R n ψ R n z d : R R n y R n z R n u R n γ R n λ R n ψ R n u The variables yzu are called the differential variables and the variables γλψ are called the algebraic variables The latter often correspond to Lagrange multipliers when the DAEs are derived from some constrained variational principle [713] The initial values y 0 z 0 u 0 γ 0 ψ 0 λ 0 at t 0 are supposed to be given Some differentiability conditions on the above functions and consistency of the initial values are assumed to ensure existence and uniqueness of the solution In a neighborhood of the solution the following conditions are supposed to be satisfied q y is invertible p z is invertible 2a 2b

4 174 LO Jay / Iterative solution of SPARK methods c u is invertible m γ is invertible p z f λ f ψ h z O O is invertible r q v z O O Notice that from 1g g y = r q q y holds hence r q v z = g y q y 1 v z in 2e Under conditions 2a 2c explicit expressions for the derivatives of the differential variables can be obtained The differential equations 1a 1c lead to q t t y + q y t y dy dt 2c 2d 2e = vtyz 3a p t tyz+p y tyz dy dt + p ztyz dz dt = ftyzuγλψ 3b c t tyzu+c y tyzu dy dt + c ztyzu dz dt + c utyzu du dt = dtyzuγλψ 3c For example 3a leads to dy dt = q y t y 1 vtyz qt t y Implicit expressions for the algebraic variables can be obtained by application of the implicit function theorem From 1d 2d the algebraic variables γ can be implicitly expressed as γ = γtyzu Differentiating the constraints 1e once gives 0 = d dt ht y z = h ttyz+h y tyz dy dt + h ztyz dz dt Differentiating the constraints 1g twice leads successively to 4a 0 = d dt gty = r t tqty + rq tqty vtyz 4b 0 = d2 dt gty = r 2 tt tqty + 2rtq tqty vtyz + r qq tqty vt y zvtyz +rq tqty vt tyz +r q tqty vy tyz dy dt + r q tqty vz tyz dz dt Hence from 2e 3b 4a 4c implicit expressions for the algebraic variables λ ψ can be obtained The exact solution must satisfy these additional so-called underlying constraints 4 To be consistent the initial values y 0 z 0 u 0 γ 0 ψ 0 λ 0 at t 0 must satisfy the whole set of constraints 1d 1f 4 After one more differentiation of the constraints 1d 4a 4c explicit expressions for the derivatives of the algebraic variables can be obtained forming together with 1a 1c an underlying system of ODEs 4c

5 LO Jay / Iterative solution of SPARK methods 175 The overdetermined system of DAEs 1 4 is therefore of differential and perturbation index 1 [8] The constraints 1d 1f are often called the index constraints respectively although the notion of index especially of perturbation index is more closely related to variables than to equations [8 p 10] DAEs of perturbation index less than or equal to 1 such as 1 4 are well-posed contrary to higher perturbation index DAEs such as 1 From a numerical and computational perspective and from the mathematical point of view of well- and ill-posedness the perturbation index is certainly one of the most relevant notions of index With the equations of mechanical systems in mind where different types of forces are present see [791617] decompositions of the right-hand sides of 1a 1c can be considered vtyz = v m tyz m=1 ftyzuγλψ= f m tyzuγλψ m=1 dtyzuγλψ = d m tyzuγλψ m=1 5a 5b 5c The functions v m f m d m are supposed to have distinct properties and can therefore be numerically treated in a different way The value of m corresponds to different classes of certain types of right-hand side terms This value should be reasonably small For example mechanical systems may include different types of forces such as conservative dissipative explosive and highly oscillatory forces hence typically m = 4 For the application of the numerical methods considered in this paper the following additional assumptions are made f 1 tyzuγλψ =f 1 tyzuγ d 1 tyzuγλψ =d 1 tyzuγ 5d This is obviously not a restriction on the system 1 per se but it is used as a restriction on the application of SPARK methods see section 3 3 SPARK methods The application of SPARK methods to the overdetermined system of implicit DAEs 1 4 is tentatively given as follows generalizing the definition of [9]:

6 176 LO Jay / Iterative solution of SPARK methods Definition 1 One step of an s-stage super partitioned additive Runge Kutta SPARK method applied with stepsize h to the overdetermined system of implicit differential algebraic equations 1 4 satisfying the assumptions with decompositions 5 reads Q i q 0 h P i p 0 h C i c 0 h a ijm v m T j Y j Z j =0 for i = 1s 6a a ijm f m T j Y j Z j U j Ɣ j j j =0 a ijm d m T j Y j Z j U j Ɣ j j j =0 for i = 1s for i = 1s mt i Y i Z i U i Ɣ i =0 for i = 1s 6d ht i Y i Z i =0 for i = 1s 6e r T i q 0 +h a ij1 vt j Y j Z j =0 for i = 1s 6f q 1 q 0 h p 1 p 0 h c 1 c 0 h j=1 b j vt j Y j Z j =0 j=1 b j ft j Y j Z j U j Ɣ j j j =0 j=1 b j dt j Y j Z j U j Ɣ j j j =0 j=1 mt 1 y 1 z 1 u 1 γ 1 =0 ht 1 y 1 z 1 =0 rt 1 q 1 =0 r t t 1 q 1 +r q t 1 q 1 vt 1 y 1 z 1 =0 6b 6c 6g 6h 6i 6j 6k 6l 6m where q 0 := qt 0 y 0 p 0 := pt 0 y 0 z 0 c 0 := ct 0 y 0 z 0 u 0 T i := t 0 + c i h Q i := qt i Y i for i = 1s for i = 1s

7 LO Jay / Iterative solution of SPARK methods 177 P i := pt i Y i Z i C i := ct i Y i Z i U i t 1 := t 0 + h q 1 := qt 1 y 1 p 1 := pt 1 y 1 z 1 c 1 := ct 1 y 1 z 1 u 1 for i = 1s for i = 1s The RK coefficients matrices of m Runge Kutta RK methods based on the same quadrature formula b i c i i=1s are denoted by A m := a ijm ij=1s for m = 1 m To ensure existence and uniqueness of the numerical solution only a certain linear combination of equations 6e 6k is actually considered see equations 14e From this tentative definition of SPARK methods results a system of nonlinear equations to be solved for the internal stages Y i Z i U i Ɣ i i i for i = 1s and for the numerical approximation at t 1 given by y 1 z 1 u 1 γ 1 In general existence and uniqueness to these nonlinear equations cannot be shown unless some assumptions on the SPARK coefficients are made Also equations 6e 6k for the constraints 1e cannot be all satisfied The actual definition of SPARK methods is given in definition 5 Accurate values for the algebraic variables γ 1 λ 1 ψ 1 are not necessary for the step by step integration In any case the accuracy of these algebraic variables does not influence the convergence of the other variables and the properties of SPARK methods since the values γ 0 λ 0 ψ 0 do not enter explicitly the definition of SPARK methods For SPARK methods satisfying c s = 1 the approximations given by γ 1 := Ɣ s λ 1 := s ψ 1 := s are adequate 31 Properties of SPARK coefficients In this article I m denotes the m m identity matrix O mn denotes the m n zero matrix b := b 1 b s T is the weight vector e i := T is the ith s-dimensional unit basis vector and 0 s := 00 T is the s-dimensional zero vector It is assumed that the number of internal stages satisfies s 2 SPARK methods 6 satisfying the following assumptions are considered e T 1 A 1 = 0 T s e T s A 1 = b T es T A 3 = b T 0 T A 1 A m = s for m = 2m N N is invertible b T A 3 is invertible 7a 7b 7c 7d 7e 7f

8 178 LO Jay / Iterative solution of SPARK methods These assumptions are satisfied for example by the Lobatto family with m = 5and A 1 A 2 A 3 A 4 A 5 being the RK matrices of Lobatto IIIA-B-C-C -D coefficients respectively [9] The assumptions 7b 7c are stiff accuracy conditions Notice that 7a is also a direct consequence of 7f and 7d for m = 3 Let à 1 be the s 1 s submatrix of A 1 given by the relation 0 T s A 1 = 8 The following lemmas will be extremely useful to obtain an efficient implementation of SPARK methods applied to DAEs à 1 Lemma 2 The relation follows from 7c 7f 1 Ã1 N = A 3 9 e T s b T Proof The relation à 1 A 3 = N follows from 7d for m = 3 Hence together with 7c it leads to Ã1 N A 3 = e T s The invertibility of the left matrix on the left-hand side follows from the conditions 7e 7f Under the assumptions of lemma 2 the following s s + 1 matrices are defined 1 Ã1 Ã1 0 s 1 Q ss+1 := 0 T = I s 0 s + Pss+1 10a s 1 e T s b T where P ss+1 := 1 Ã1 Oss 1 e s e s = Oss 1 p s p s 10b e T s p s := 1 Ã1 e s = e T s vs c Lemma 3 From 7c 7f it follows that Am 0 Q s ss+1 b T = A s for m = 2m

9 LO Jay / Iterative solution of SPARK methods 179 Moreover if in addition 7b holds then the following relation is obtained A1 0 Q s ss+1 b T = A s Proof The relations 1 Ã1 Ã1 0 s 1 Am 0 s = A 0 T s 1 b T 3 0 s 0 e T s follow directly from 7d and lemma 2 For m = 1 the relation 1 Ã1 Ã1 0 s 1 A1 0 s = A 0 T s 1 b T 1 0 s 0 follows from which is a simple consequence of 7b e T s A 1 0 s Oss 1 e s e s b T = O ss+1 0 for m = 2m Lemma 4 Consider the matrix Q ss+1 as defined in 10 and the s + 1-dimensional vector q s+1 := T Then the following s + 1 s + 1 matrix Q s+1s+1 := is invertible Qss+1 q T s+1 11 Proof The expressions 10 give Is 1 v s 1 v s 1 Q ss+1 = 12 0 T s hence detq s+1s+1 = 1 is easily obtained In fact the inverse to Q s+1s+1 can be given explicitly by using the factorization Q s+1s+1 = R s+1s+1 S s+1s+1 where 1 O 1 O R s+1s+1 := ps 1 S s+1s+1 := 1 O 1 O a 13b

10 180 LO Jay / Iterative solution of SPARK methods Hence Q 1 s+1s+1 = S 1 s+1s+1 R 1 s+1s+1 holds where 1 O S 1 s+1s+1 = 1 O 1 1 R 1 s+1s+1 = 1 O ps 1 O 1 32 The system of nonlinear equations It is essential to put the system of nonlinear equations in a form such that preconditioning the linear equations of the modified Newton method can be done in a straightforward manner following the results of [10 12] see section 5 From the assumption 7a and the consistency condition rt 0 q 0 = 0the equation 6f for i = 1 is automatically satisfied A consequence of the assumption 7b is rt 1 q 1 =r T s q 0 +h a sj1 vt j Y j Z j j=1 therefore by 6f for i = s equation 6l is also automatically satisfied Instead of solving directly the remaining set of equations of 6 some specific linear transformations are applied The remaining equations are expressed by making use of the matrices Q ss+1 10 and Q s+1s+1 11 as follows: Definition 5 The actual definition of SPARK methods applied to is taken as m Q 1 q 0 h a 1jm v m T j Y j Z j Q s+1s+1 I ny =0 14a Q s q 0 h a sjm v m T j Y j Z j q 1 q 0 h b j vt j Y j Z j j=1

11 LO Jay / Iterative solution of SPARK methods 181 Q s+1s+1 I nz m P 1 p 0 h a 1jm f m T j Y j Z j U j Ɣ j j j =0 14b P s p 0 h a sjm f m T j Y j Z j U j Ɣ j j j p 1 p 0 h b j ft j Y j Z j U j Ɣ j j j j=1 Q s+1s+1 I nu m C 1 c 0 h a 1jm d m T j Y j Z j U j Ɣ j j j =0 14c C s c 0 h a sjm d m T j Y j Z j U j Ɣ j j j c 1 c 0 h b j dt j Y j Z j U j Ɣ j j j j=1 mt 1 Y 1 Z 1 U 1 Ɣ 1 Q s+1s+1 I nγ =0 14d mt s Y s Z s U s Ɣ s mt 1 y 1 z 1 u 1 γ 1 ht 1 Y 1 Z 1 Q ss+1 I nλ ht s Y s Z s =0 14e ht 1 y 1 z 1 1 h r T 2 q 0 +h a 2j1 vt j Y j Z j j=1 1 Ã1 es T I nψ =0 14f 1 h r T s q 0 +h a sj1 vt j Y j Z j j=1 r t t 1 q 1 +r q t 1 q 1 vt 1 y 1 z 1

12 182 LO Jay / Iterative solution of SPARK methods Equations 14e correspond only to a linear combination of the constraints 6e 6k From 12 equation 6k is truly the only one preserved among 6e 6k this implies that the numerical solution at t 1 still satisfies the constraint 1e This linear combination 14e of 6e 6k is somehow necessary to ensure existence and uniqueness of the numerical solution since in 6 there are only s algebraic variables i i = 1s for s + 1 sets of equations 6e 6k In order to combine the constraints 6f and 6l properly the equations 6f have been multiplied by 1/h where h is the stepsize Since rt 0 q 0 = 0 this can be interpreted as a finite difference approximation to drtqty/dt = 0atT i d dt r T i q T i yt i rt iq 0 +h s j=1 a ij1vt j Y j Z j rt 0 q 0 h The main reason for these linear transformations is to obtain an advantageous structure of the approximate Jacobian given in section 4 for the construction of efficient preconditioners to be discussed in section 5 Notice that because of the factorization 13a 13b multiplication by Q s+1s+1 of equations 14a 14c has an implication easily interpretable For example 14b can be rewritten as Rs+1s+1 I nz m P 1 p 0 h a 1jm f m T j Y j Z j U j Ɣ j j j =0 14g P s p 0 h a sjm f m T j Y j Z j U j Ɣ j j j p 1 P s h b j a sjm f m T j Y j Z j U j Ɣ j j j One effect is to remove stiffness from the last set of equations provided the terms causing stiffness are treated with coefficients a ijm satisfying the stiff accuracy condition a sjm = b j for j = 1ssee 7b 7c A set of dummy equations can be appended to 14e 14f ht 1 y 1 z 1 ht s Y s Z s +λ 1 =0 14h r t t 1 q 1 +r q t 1 q 1 vt 1 y 1 z 1 r t T s Q s +r q T s Q s vt s Y s Z s +ψ 1 =0 14i These equations must be taken as a dummy definition of λ 1 ψ 1 Of course the exact solution λt ψt at t = t 0 + h must not be approximated by λ 1 ψ 1 as given above but for example by s s when c s = 1 The unknowns λ 1 ψ 1 have been included only for ease of presentation in the derivation hereafter but they are actually not unknowns

13 LO Jay / Iterative solution of SPARK methods 183 of the nonlinear system 14a 14f In 14 a system of nonlinear equations has been obtained for the following vector collecting all unknowns X := Y 1 Z 1 U 1 Ɣ Y s Z s U s Ɣ s s s y 1 z 1 u 1 γ 1 λ 1 ψ 1 T 15 4 Inexact modified Newton iterations for SPARK methods The iteration schemes proposed to solve the system of nonlinear equations corresponding to the application of IRK methods to DAEs have generally been based on ad hoc modifications of the simplified Newton method [5] For SPARK methods there are additional difficulties to obtain an efficient implementation due to their additive and partitioned nature This paper proposes the use of inexact modified Newton iterations or more precisely using another terminology of modified Newton-iterative methods extending techniques developed in [10 12] Instead of solving exactly the linear systems of the modified Newton method they can be solved approximately and iteratively after application of specific linear transformations and the use of a preconditioner 41 Inexact modified Newton iterations Modified Newton iterations applied to the set of equations 14 read as follows M X k = G X k X k+1 = X k + X k k = where M is a modified Jacobian ie roughly speaking an approximation to the exact Jacobian X := X 1 X s x 1 T 17a X i := Y i Z i U i Ɣ i i i T for i = 1s 17b x 1 := y 1 z 1 u 1 γ 1 λ 1 ψ 1 T 17c and GX corresponds to the expressions in 14 reordered accordingly A direct decomposition of the modified Jacobian M may be inefficient In the inexact modified Newton method the linear systems 16 are solved only approximately generally by a preconditioned linear iterative method such as preconditioned versions of Richardson or GMRES iterations [ ] This requires the construction of a good preconditioner see section 5 A sequence of iterates X k with a residual error r k := M X k + G X k is obtained at each iteration Sufficient a priori and a posteriori conditions to ensure convergence of the inexact modified Newton iterates toward the solution of a system of nonlinear equations have been given in [11] In combination with a linear iterative method the inexact Newton method is called a modified Newtoniterative method The use of preconditioned linear iterative methods for the numerical solution of ODEs and DAEs was first considered in the context of implicit multistep methods by Brown et al [2]

14 184 LO Jay / Iterative solution of SPARK methods 42 The simplified Jacobian In a standard approach the system of nonlinear equations 14 is solved by simplified Newton iterations This requires the simplified Jacobian which is the Jacobian at the initial guess The simplified Jacobian corresponding to the equations 14a 14f with respect to the variables in 15 can be expressed as follows Q s+1s+1 q y O O O O O Am 0 s h Q s+1s+1 b T 0 m=1 Q s+1s+1 p y p z O O O O Am 0 s h Q s+1s+1 b T 0 m=1 Q s+1s+1 c y c z c u O O O Am 0 s h Q s+1s+1 b T 0 m=1 v my v mz O O O O 18a f my f mz f mu f mγ f mλ f mψ 18b d my d mz d mu d mγ d mλ d mψ 18c Q s+1s+1 m y m z m u m γ O O 18d Q ss+1 h y h z O O O O 18e 1 Ã1 Ã1 0 s 1 es T 0 T r q v y r q v z O O O O s 1 1Os 1s Ã1 0 s 1 + es T 0 T r tq q y + r qq q y v O O O O O s 1 = Q ss+1 r q v y r q v z O O O O + O ss p s rtq q y + r qq q y v O O O O O 18f where the symbol denotes the tensor product The arguments of the expressions q y v my etc which have been omitted are given by the initial values t 0 y 0 z 0 u 0 γ 0 λ 0 ψ 0 Strictly speaking this is in fact not really the simplified Jacobian since the initial guess of the iterations is generally not given by the initial values Nevertheless the terminology of simplified Jacobian to refer to 18 will be kept since this is how it is generally called following eg [6 section IV8] The terminology of modified Jacobian would actually be more correct 43 An approximate Jacobian Some modifications to the simplified Jacobian 18 can actually be used since it is not necessary to keep the full simplified Jacobian to ensure convergence of the modified

15 LO Jay / Iterative solution of SPARK methods 185 Newton iterates 16 The expression of a so-called approximate Jacobian L is given here It will be used in the discussion given in section 5 about the construction of preconditioners It should be noticed that it is not necessary to use exactly the approximate Jacobian presented here to ensure convergence of the whole inexact modified Newton procedure additional modifications are possible A main point in the specification of the approximate Jacobian concerns the equations q 1 Q s h p 1 P s h c 1 C s h b j a sjm v m T j Y j Z j =0 19a b j a sjm f m T j Y j Z j U j Ɣ j j j =0 19b b j a sjm d m T j Y j Z j U j Ɣ j j j =0 19c mt 1 y 1 z 1 u 1 γ 1 mt s Y s Z s U s Ɣ s =0 ht 1 y 1 z 1 ht s Y s Z s +λ 1 =0 r t t 1 q 1 +r q t 1 q 1 vt 1 y 1 z 1 r t T s Q s +r q T s Q s vt s Y s Z s +ψ 1 =0 19d 19e 19f which involve the numerical solution at the endpoint t 1 see 14 Denoting minus the left-hand side of equations 19 by r 1 one step of the simplified Newton method for these equations reads where E + F x 1 X s + J d x 1 h b T es T A m Jm X = r 1 20 m=1 with X i for i = 1s and x 1 as in 17 and X := X 1 X s T 21 q y O O O O O p y p z O O O O E := c y c z c u O O O m y m z m u m γ O O h y h z O O O O r q v y r q v z O O O O

16 186 LO Jay / Iterative solution of SPARK methods F := rtq q y + r qq q y v O O O O O v my v mz O O O O f my f mz f mu f mγ f mλ f mψ J m := d my d mz d mu d mγ d mλ d mψ J d := O O O O I O O O O O O I Thequantities y 1 z 1 u 1 γ 1 and Y s Z s U s Ɣ s both approximate the exact solution yt zt ut γt at t = t 0 + h = t 1 = T s By 19a 19d they are close to each other provided stiff terms are treated by stiffly accurate RK coefficients Hence some terms in 20 can be neglected leading to some sort of fixed-point iterations for y 1 z 1 u 1 γ 1 Since the variables λ 1 ψ 1 are dummy variables which are directly defined by 14h 14i they should have no influence on the numerical solution to the system of equations 14a 14f Therefore in 20 the term J d x 1 and the components of r 1 corresponding to 19e 19f can be neglected The dummy values λ 1 ψ 1 can be set explicitly after each modified Newton iteration such that the equations 14h 14i are satisfied exactly for the current iterate X k This means that the corresponding components of r 1 can be assumed to vanish Provided stiff terms are treated by stiffly accurate coefficients ie satisfying a sjm = b j for j = 1s the linear equation 20 can be simplified for example to E X s + E x 1 = r 1 22 From 18 the whole set of equations 14 and variables can be reordered according to 15 such that after application of lemma 3 the corresponding approximate Jacobian L considered here is expressed as follows

17 L := Q s+1s+1 E h LO Jay / Iterative solution of SPARK methods 187 A1 0 s A3 0 0 T J s 0 1 h s 0 T J s J 23 where from 5d O O O O O O O O O O f mλ f mψ O O O O f λ f ψ J 0 := O O O O d mλ d mψ m=2 = O O O O d λ d ψ 24a v 1y v 1z O O O O f 1y f 1z f 1u f 1γ O O J 1 := d 1y d 1z d 1u d 1γ O O 24b v my v mz O O O O f my f mz f mu f mγ O O J := d my d mz d mu d mγ O O m=2 24c The particular tensorial structure of the approximate Jacobian L can now be used to obtain good preconditioners It will be discussed in section 5 It would not be possible to do so if the simplified Jacobian 18 was kept Some other parts of the approximate Jacobian such as h y and r q v y can actually be neglected without jeopardizing convergence of the inexact modified Newton iterates 5 Preconditioning the approximate Jacobian In this section the discussion is based on the approximate Jacobian L 23 though as previously mentioned other choices are possible Consider a linear system with matrix L X as in 17 and right-hand side R decomposed as L X = R 25 R = R 1 R s r 1 T

18 188 LO Jay / Iterative solution of SPARK methods 51 Transforming the linear system As a first step the relation 22 can be introduced into the first s subequations of 25 A reduced linear system L X = R 26 is obtained for X given in 21 where L := I s E ha 1 J 1 ha 3 J 0 + J R := R 1 R s T p s r 1 by using the decomposition 13 for Q s+1s+1 in 23 As a second step the quantities i and i for i = 1s can be replaced in 17 as follows 1 = h 1 A 3 I nλ s s Hence a linear system is obtained with matrix where 1 = h 1 A 3 I nψ s s K x = b 27 K = I s E J 0 ha 1 J 1 ha 3 J 28 q y O O O O O p y p z O O f λ f ψ E J 0 = c y c z c u O d λ d ψ m y m z m u m γ O O h y h z O O O O r q v y r q v z O O O O Under the assumptions 2 the matrix E J 0 is invertible For nonstiff DAEs the terms ha 1 J 1 and ha 3 J in 28 can be neglected Once an approximation to the solution of the linear system 26 is obtained it remains to define an approximation to y 1 z 1 u 1 γ 1 The relation 20 or 22 can be used for that purpose The current iterates for λ 1 ψ 1 can be defined directly from 14h 14i but this need not be done explicitly The linear system 25 with approximate Jacobian L 23 has been transformed under a form such that preconditioning the linear equations 27 can now be done in a straightforward manner following the results of [10 12] The preconditioner developed in [1011] will not be described here The main drawback of this preconditioner is the

19 LO Jay / Iterative solution of SPARK methods 189 fact that although its decomposition is parallelizable the solution of the linear systems involved is not Instead a new truly parallel preconditioner is presented in the next subsection 52 A truly parallel preconditioner In this subsection some recent results of [12] are followed and briefly presented The linear system 27 is solved approximately by application of linear iterative methods with a preconditioner Q K 1 of the form Q := H 1 G H 1 where H := I s E J 0 hɣ 1 J 1 hɣ 3 J G := I s E J 0 h 1 J 1 h 3 J The coefficients matrices Ɣ 1 and Ɣ 3 are chosen to be diagonal Ɣ 1 := diagγ 11 γ 21 γ s1 Ɣ 3 := diagγ 13 γ 23 γ s3 with γ 11 = 0 because of 7a leading to where H 1 = O H 1 1 O H 1 2 H i := E J 0 hγ i1 J 1 hγ i3 J H 1 s for i = 1s The matrices H i can be decomposed independently hence in parallel Solving a linear system with matrix H can also be done in parallel since it is block-diagonal This is the main advantage of this preconditioner compared to the one presented in [1011] The coefficients of Ɣ 1 Ɣ 3 1 and 3 still remain to be fixed to some values Assuming the coefficients of Ɣ 1 and of Ɣ 3 to be given the coefficients of 1 and 3 are taken as where 1 = Ɣ 1  T s  11 Ɣ 1  1 1 Ɣ 1 3 = Ɣ 3 A 1 3 Ɣ 3 à 1 =  11  1 Ɣ 1 := diagγ 21 γ s1 with à 1 as in 8 The coefficients matrices 1 and 3 have been separately determined such that the preconditioner Q is asymptotically correct when considering the Dahlquist test equation y = λy Reλ 0

20 190 LO Jay / Iterative solution of SPARK methods The coefficients γ i1 for i = 2s and γ i3 for i = 1s are free They are required to satisfy γ 1i > 0fori =2s and γ i3 > 0fori =1s which is a natural assumption to ensure the invertibility of the matrices H i These coefficients can be chosen for example to minimize λ i Mz 1 Rez 0 i=1s where z = hλ Mz = Qz Kzandλ i Mz for i = 1s are the s eigenvalues of Mz Whenγ i1 = γ 1 for i = 2s and γ i3 = γ 3 for i = 1s only one or two matrix decompositions beside E J 0 are needed This can be quite advantageous on a serial computer The cost of computing a matrix vector product Qv with at least s processors on a parallel computer consists of one decomposition of H i on each processor two linear systems with matrix H i to be solved one local matrix vector product with each matrix E J 0 hj 1 andhj and some communication between processors according to the nonzero elements of the coefficients matrices 1 and 3 6 Conclusion The approximation of a certain class of DAEs by SPARK methods has been considered The main difficulty of these methods resides in finding a way to implement them efficiently Certain linear transformations are applied to the resulting system of nonlinear equations such that an efficient preconditioner to the linear systems of the modified Newton method can be constructed These linear transformations rely heavily on specific properties of the SPARK coefficients An approximate Jacobian of the system of nonlinear equations is presented Based on this approximate Jacobian a new truly parallelizable preconditioner is given References [1] KE Brenan SL Campbell and LR Petzold Numerical Solution of Initial-Value Problems in Differential Algebraic Equations SIAM Classics in Applied Mathematics 2nd ed SIAM Philadelphia PA 1996 [2] PN Brown AC Hindmarsh and LR Petzold Using Krylov methods in the solution of large-scale differential algebraic systems SIAM J Sci Comput [3] G Golub and CF van Loan Matrix Computations Johns Hopkins Studies in the Mathematical Science 3rd ed Johns Hopkins Univ Press Baltimore/London 1996 [4] A Greenbaum Iterative Methods for Solving Linear Systems Frontiers in Applied Mathematics Vol 17 SIAM Philadelphia PA 1997 [5] E Hairer Ch Lubich and M Roche The Numerical Solution of Differential Algebraic Systems by Runge Kutta Methods Lecture Notes in Mathematics Vol 1409 Springer Berlin 1989 [6] E Hairer and G Wanner Solving Ordinary Differential Equations II Stiff and Differential Algebraic Problems Comput Math Vol 14 2nd revised ed Springer Berlin 1996 [7] EJ Haug Computer Aided Kinematics and Dynamics of Mechanical Systems Vol I: Basic Methods Allyn and Bacon Boston USA 1989

21 LO Jay / Iterative solution of SPARK methods 191 [8] LO Jay Runge Kutta type methods for index three differential algebraic equations with applications to Hamiltonian systems PhD thesis Department of Mathematics University of Geneva Switzerland 1994 [9] LO Jay Structure preservation for constrained dynamics with super partitioned additive Runge Kutta methods SIAM J Sci Comput [10] LO Jay A parallelizable preconditioner for the iterative solution of implicit Runge Kutta type methods J Comput Appl Math [11] LO Jay Inexact simplified Newton iterations for implicit Runge Kutta methods SIAM J Numer Anal [12] LO Jay Preconditioning and parallel implementation of implicit Runge Kutta methods Technical Report Dept of Math University of Iowa USA 2001 in preparation [13] PJ Rabier and WC Rheinboldt Nonholonomic Motion of Rigid Mechanical Systems from a DAE Viewpoint SIAM Philadelphia PA 2000 [14] Y Saad Iterative Methods for Sparse Linear Systems PWS Boston MA 1996 [15] Y Saad and MH Schultz GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems SIAM J Sci Statist Comput [16] WO Schiehlen ed Multibody Systems Handbook Springer Berlin 1990 [17] WO Schiehlen ed Advanced Multibody System Dynamics Simulation and Software Tools Kluwer Academic London 1993